1001.1184

7/27/2019 1001.1184

1/12

a r X i v : 1 0 0 1 . 1 1 8 4 v 1 [ q - f i n . P R ] 8 J a n 2 0 1 0

STOCHASTIC DISCOUNT FACTORS

CONSTANTINOS KARDARAS

Abstract. The valuation process that economic agents undergo for investments with uncertainpayoff typically depends on their statistical views on possible future outcomes, their attitudes towardrisk, and, of course, the payoff structure itself. Yields vary across different investment opportunitiesand their interrelations are difficult to explain. For the same agent, a different discounting factorhas to be used for every separate valuation occasion. If, however, one is ready to accept discountingthat varies randomly with the possible outcomes, and therefore accepts the concept of a stochasticdiscount factor, then an economically consistent theory can be developed. Asset valuation becomes amatter of randomly discounting payoffs under different states of nature and weighing them accordingto the agents probability structure. The advantages of this approach are obvious, since a singlediscounting mechanism suffices to describe how any asset is priced by the agent.

0. Introduction

Within active and liquid nancial markets, economic agents are able to make investment de-cisions. Capital is allocated today in exchange for some future income stream. If there is nouncertainty regarding the future payoff of an investment opportunity, the yield that will be askedon the investment will equal the risk-free interest rate prevailing for the time period covering thetime of investment until the time of the payoff. However, in the presence of any payoff uncertaintywhen undertaking an investment venture, economic agents will typically ask for risk compensation ,and thus for some investment-specic yield, which will discount the expected future payoff stream.The yields, that particular agents ask for, depend both on their statistical views on possible futureoutcomes, as well as their attitudes towards risk.

Yields vary across different investment opportunities and their interrelations are difficult to ex-plain. For the same agent, a different discounting factor has to be used for every separate valuationoccasion. If one, however, is ready to accept discounting that varies randomly with the possible out-comes, and therefore accepts the concept of a stochastic discount factor, then a very economicallyconsistent theory can be developed. Asset valuation becomes a matter of randomly discountingpayoffs under different states of nature and weighing them according to the agents probabilitystructure. The advantages of this approach are obvious, since a single discounting mechanismsuffices to describe how any asset is priced by the agent.

Date : January 8, 2010.Key words and phrases. Arbitrage, asset pricing, (local) martingales, risk premium, state-price densities, utility

indifference pricing.1
http://arxiv.org/abs/1001.1184v1http://arxiv.org/abs/1001.1184v1http://arxiv.org/abs/1001.1184v1http://arxiv.org/abs/1001.1184v1http://arxiv.org/abs/1001.1184v1http://arxiv.org/abs/1001.1184v1http://arxiv.org/abs/1001.1184v1http://arxiv.org/abs/1001.1184v1http://arxiv.org/abs/1001.1184v1http://arxiv.org/abs/1001.1184v1http://arxiv.org/abs/1001.1184v1http://arxiv.org/abs/1001.1184v1http://arxiv.org/abs/1001.1184v1http://arxiv.org/abs/1001.1184v1http://arxiv.org/abs/1001.1184v1http://arxiv.org/abs/1001.1184v1http://arxiv.org/abs/1001.1184v1http://arxiv.org/abs/1001.1184v1http://arxiv.org/abs/1001.1184v1http://arxiv.org/abs/1001.1184v1http://arxiv.org/abs/1001.1184v1http://arxiv.org/abs/1001.1184v1http://arxiv.org/abs/1001.1184v1http://arxiv.org/abs/1001.1184v1http://arxiv.org/abs/1001.1184v1http://arxiv.org/abs/1001.1184v1http://arxiv.org/abs/1001.1184v1http://arxiv.org/abs/1001.1184v1http://arxiv.org/abs/1001.1184v1http://arxiv.org/abs/1001.1184v1http://arxiv.org/abs/1001.1184v1http://arxiv.org/abs/1001.1184v1http://arxiv.org/abs/1001.1184v1http://arxiv.org/abs/1001.1184v1http://arxiv.org/abs/1001.1184v1http://arxiv.org/abs/1001.1184v1

7/27/2019 1001.1184

2/12

7/27/2019 1001.1184

3/12

STOCHASTIC DISCOUNT FACTORS 3

is invested in the baseline asset. If X (x, ) is the wealth generated starting from capital x andinvesting according to , then X (x, )0 = x and

(1.1) X (x ; )T = x d

i=1i S i0

S 0T S 00

+d

i=1i S iT = x

S 0T S 00

+d

i=1i S iT

S 0T S 00

S i0 ,

or in deated terms T X (x ; )T = x + di=1 i T S iT S i0 . The agents objective is to choose astrategy in such a way as to maximize expected utility , i.e., nd such that

(1.2) E P U X (x ; )T = supR d

E P U X (x ; )T .

The above problem will indeed have a solution if and only if no arbitrages exist in the market.By denition, an arbitrage is a wealth generated by some R d such that P [X (x ; )T 0] = 1and P [X (x ; )T > 0] > 0. It is easy to see that arbitrages exist in the market if and only if sup R d E P [U (X

(x ; )T )] is not attained by some R

d . Assuming then the No-Arbitrage (NA)condition, concavity of the function R d E P [U (X (x ; )T )] will imply that the rst-order condi-

tions i =

E P U X (x ; )T = 0 , for all i = 1 , . . . , d ,

will provide the solution to the problem. Since the expectation is just a nite sum, the differentialoperator can pass inside, and then the rst-order conditions for optimality are

(1.3) 0 = E P

i = U X (x ; )T = E

P U X (x ; )T S iT

S 0T S 00

S i0 , i = 1 , . . . , d .

The above is a non-linear system of d equations to be solved for d unknowns ( 1, . . . , d). Under(NA), the system ( 1.3) has a solution . Actually, under a trivial non-degeneracy condition in themarket the solution is unique; even if the optimal strategy is not unique, strict concavity of U implies that the optimal wealth X (x ; )T generated is unique.

A little bit of algebra on ( 1.3) gives, for all i = 1 , . . . , d ,

(1.4) S i0 = EP Y T S iT , where Y T :=

U X (x ; )T

E P (S 0T /S 00 )U X (x ; )T

Observe that since U is continuously differentiable and strictly increasing, U is a strictly positivefunction, and therefore P [Y T > 0] = 1. Also, (1.4) trivially also holds for i = 0. Note that therandom variable Y T that was obtained above depends on the utility function U , the probability P ,as well as on the initial capital x R .

Denition 1.1. In the model described above, a process Y = ( Y t )t=0 ,T will be called a stochastic discount factor if P [Y 0 = 1 , Y T > 0] = 1 and S i0 = E

P Y T S iT for all i = 0 , . . . , d .

If Y is a stochastic discount factor, using ( 1.1) one can actually show that

(1.5) E P Y T X (x ; )T = x, for all x R and R

d .

In other words, the process Y X (x ; ) is a P -martingale for all x R and R d .

7/27/2019 1001.1184

4/12

4 CONSTANTINOS KARDARAS

1.3. Connection with risk-neutral valuation. Since E P [S 0T Y T ] = S 00 > 0, we can dene aprobability mass Q by requiring that Q () = S 0T ()/S 00 Y T ()P (), which denes a probability Qon subsets of in the obvious way. Observe that, for any A , Q [A] > 0 if and only if P [A] > 0;we say that the probabilities P and Q are equivalent and we denote by Q P . Now, rewrite ( 1.4)as

(1.6) S i0 = EQ T S iT , for all i = 0 , . . . , d .

A probability Q , equivalent to P , with the property prescribed in ( 1.6) is called risk-neutral or an equivalent martingale measure . In this simple framework, stochastic discount factors andrisk-neutral probabilities are in one-to-one correspondence. In fact, more can be said.

Theorem 1.2 (Fundamental Theorem of Asset Pricing) . In the discrete model as described previ-ously, the following three conditions are equivalent.

(1) There are no arbitrage opportunities.(2) A stochastic discount factor exists.(3) A risk neutral probability measure exists.

The Fundamental Theorem of Asset Pricing was rst formulated in [ 10] and it took twenty yearsto reach a very general version of it in general semimartingale models that are beyond the scope of our treatment here. The interested reader can check the monograph [ 3], where the history of theTheorem and all proofs is presented.

1.4. The important case of the logarithm. The most well-studied case of utility on the realline is U (x) = log( x), both because of its computational simplicity and for the theoretical valuethat it has. Since the logarithmic function is only dened on the strictly positive real line, it doesnot completely fall in the aforementioned framework, but it is easy to see that the described theoryis still valid.

Consider an economic agent with logarithmic utility that starts with initial capital x = 1. CallX = X (1; ) the optimal wealth corresponding to log-utility maximization. The fact that U (x) =1/x allows to dene a stochastic discount factor Y via Y 0 = 1 and

Y

T =1

X T E P 1/ ( T X

T )

From E P [Y T X

T ] = 1 it follows that EP [1/ ( T X T )] = 1 and therefore Y

= 1 /X . This simplerelationship between the log-optimal wealth and the stochastic discount factor that is induced by itwill be one of the keys to characterize existence of stochastic discount factors in more complicatedmodels and their relationship with absence of free lunches. It will nd good use in the next Section2 for the case of models using Ito processes.

7/27/2019 1001.1184

5/12


1.5. Arbitrage-free prices. For a claim with random payoff H T at time T , an arbitrage-free price H 0 is a price at time zero such that the extended market that consists of the original tradedassets with asset prices S i , i = 0 , . . . , d , augmented by the new claim, remains arbitrage-free. If the claim is perfectly replicable , i.e., if there exists x R and R d such that X (x ; )T = H T , itis easily seen that the unique arbitrage-free price for the claim is x. However, it is frequently thecase that a newly-introduced claim in not perfectly replicable using the existing liquid assets. Inthat case, there exist more than one arbitrage-free price for the claim; actually, the set of all thepossible arbitrage-free prices is {E P [Y T H T ] | Y is a stochastic discount factor }. To see this, rstpick a stochastic discount factor Y T and set H 0 = E [Y T H T ]; then, Y remains a stochastic discountfactor for the extended market, which therefore does not allow for any arbitrage opportunities.Conversely, if H 0 is an arbitrage-free price for the new claim, we know from Theorem 1.2 that thereexists a stochastic discount factor Y for the extended market, which satises H 0 = E [Y T H T ] andis trivially a stochastic discount factor for the original market. The result we just mentioned gives justice to the appellation Fundamental Theorem of Asset Pricing for Theorem 1.2.

1.6. Utility Indifference Pricing. Suppose that a new claim promising some random payoff attime T is issued. Depending on the claims present traded price, an economic agent might beinclined to take a long or short position this will depending on whether the agent considers themarket price low or high, respectively. There does exist a market-price-level of the claim that willmake the agent indifferent between longing or shorting an innitesimal 2 amount of asset. This pricelevel is called indifference price . In the context of claim valuation, utility indifference prices havebeen introduced in [ 2]3, but have been widely used previously in the Economics science. Indifferenceprices depend on the particular agents views, preferences, as well as portfolio structure, and shouldnot be confused with market prices, which are established using the forces of supply and demand.

Since the discussed preference structures are based on expected utility , it makes sense to try andunderstand quantitatively how utility indifference prices are formed. Under the present set-up,consider a claim with random payoff H T at time T . The question we wish to answer is: what is theindifference price H 0 of this claim today for an economic agent.

For the time being, let H 0 be any price set by the market for the claim. The agent will invest inthe risky assets and will hold units of them, as well as the new claim, taking a position of units.Then, the agents terminal payoff is

X (x ; ,)T := X (x ; )T + H T

S 0T S 00

H 0 .

2We stress innitesimal because when the portfolio holdings of the agent change, the indifference prices alsochange; thus, for large sales or buys that will considerably change the portfolio structure, there might appear incentive,that was not there before, to sell or buy the asset.

3For this reason, utility indifference prices are sometimes referred to as Davis prices.

7/27/2019 1001.1184

6/12


The agent will again maximize expected utility, i.e., will invest ( ,) R d R such that

(1.7) E P U X (x ; , )T = sup(,)R d R

E P U X (x ; ,)T .

If H 0 is the agents indifference price, it must be that = 0 in the above maximization problem;then, the agents optimal decision regarding the claim would be not to buy or sell any units of it. Inparticular, the concave function R E P U X (x ; ,)T should achieve its maximum at = 0.First-order conditions give that H 0 is the agents indifference price if

0 = =0

E P U X (x ; ,)T = EP U X (x ; ,0)T X T

S 0T S 00

X 0

A remark is in order before writing down the indifference-pricing formula. The strategy that hasbeen appearing above represents the optimal holding in the liquid traded assets when all assets and the claim are available it is not in general the agents optimal asset holdings if the claim wasnot around. Nevertheless, if the solution of problem ( 1.7) is such that the optimal holdings in theclaim are = 0, then are also the agents optimal asset holdings if there was no claim to beginwith. In other words, if = 0, X (

x ; ,0)T is exactly the same quantity X

(x ; )T that appears in ( 1.3).

Remembering the denition of the stochastic discount factor Y T of (1.4), we can write

H 0 = E P [Y T H T ]

It is important to observe that Y T depends on a lot of things: namely, the probability P , the utilityU and the initial capital x, but not on the particular claim to be valued. Thus, we need only oneevaluation of the stochastic discount factor and we can use it to nd indifference prices with respectto all sorts of different claims.

1.7. State price densities. For a xed , consider an Arrow-Debreau security that pays off aunit of account at time T if the state-of-nature is , and nothing otherwise. The indifference priceof this security for the economic agent is p() := Y ()P (). Since Y appears as the density of thestate price p with respect to the probability P , stochastic discount factors are also termed state price densities in the literature. For two states-of-the-nature and of such that Y () < Y ( ),an agent that uses the stochastic discount factor Y considers a more unfavorable state than and is inclined to pay more for insurance against adverse market movements.

1.8. Comparison with real-world valuation. Only for the purposes of what is presented here,assume that S 00 = 1 and S 0T = 1 + rT for some r R + . Let Y be a stochastic discount factor; then,

we have 1 = S 00 = E P [Y T S 0T ] = (1 + rT )E P [Y T ]. Pick then any claim with random payoff H T attime T and use H 0 = E P [Y T H T ] to write

(1.8) H 0 =1

1 + rT E P [H T ] + covP (Y T , H T ),

where cov P (, ) is used to denote covariance of two random variables with respect to P . The rstterm (1 + rT ) 1E P [H T ] of the above formula describes real world valuation for an agent whowould be neutral under his views P in facing the risk coming from the random payoff H T . This

7/27/2019 1001.1184

7/12


risk-neutral attitude is absent usually: agents require compensation for the risk they undertake, ormight even feel inclined to pay more for a security that will insure them in cases of unfavorableoutcomes. This is exactly mirrored by the correction factor cov P (Y T , H T ) appearing in ( 1.8). If thecovariance of Y T and H T is negative, the claim tends to pay more when Y T is low. By the discussionin 1.7, this means that the payoff will be high in states that are not greatly feared by the agent,who will therefore be inclined to pay less than what the real-world valuation gives. On the contrary,if the covariance of Y T and H T is positive, H T will pay off higher in dangerous states-of-nature forthe agent (where Y T is also high), and the agents indifference price will be higher than real-worldvaluation.

2. Stochastic Discount Factors for It o Processes

2.1. The model. Uncertainty is modeled via a probability space ( , F , F , P ), where F = ( F t )t[0,T ]is a ltration representing the ow of information. The market consists of a locally riskless savingsaccount whose price process S 0 satises S 00 > 0 and

dS 0tS 0t

= r t dt, t [0, T ],

for some F -adapted, positive short rate process r = ( r t )tR . It is obvious that S 0t = S 00 exp( t

0 r u du)for t [0, T ]. We dene the deator via

t =S 00S 0t

= exp t

0r u du , t [0, T ].

The movement of d risky assets will be modeled via Ito processes:

dS it

S it

= bit dt + it , dW t , t R + , i = 1 , . . . , d .

Here, b = ( b1, . . . , bd ) is the F -adapted d-dimensional process of appreciation rates , W = ( W 1, . . . , W m )is an m-dimensional P -Brownian motion representing the sources of uncertainty in the market, and

, denotes the usual inner product notation: it , dW t =m j =1

jit dW

jt where ( ji )1 j m, 1 i d

if the F -adapted ( m d)-matrix-valued process whose entry jit represents the impact of the j thsource of uncertainty on the ith asset at time t [0, T ]. With denoting transposition, c := is the d d local covariation matrix . To avoid degeneracies in the market, it is required that cthas full rank for all t [0, T ], P -almost surely. This implies in particular that d m there aremore sources of uncertainty in the market than are liquid assets to hedge away the uncertainty risk.

Model of this sort are classical in the Quantitative Finance literature see for example [8].

Denition 2.1. A risk premium is any m-dimensional, F -adapted process satisfying = b r 1 ,where 1 is the d-dimensional vector with all unit entries.

The terminology risk premium is better explained for the case d = m = 1; then = ( b r )/ isthe premium that is asked by investors for the risk associated with the (only) source of uncertainty.In the general case, j can be interpreted as the premium required for the risk associated with

7/27/2019 1001.1184

8/12


the j th source of uncertainty, represented by the Brownian motion W j . In incomplete markets,when d < m , Proposition 2.2 shows all the different choices for . Each choice will parametrize thedifferent risk attitudes of different investors. In other words, risk premia characterize the possiblestochastic discount factors, as shall be revealed in Theorem 2.7.

If m = d, the equation = b r 1 has only one solution: = c 1(b r 1 ). If d < m thereare many solutions, but they can be characterized using easy linear algebra.

Proposition 2.2. The risk premia are exactly all processes of the form = + , where :=c 1(b r 1 ) and is any adapted process with = 0 .

If = + in the notation of Proposition 2.2, then , = ( b r 1 )c 1 = 0. Then,| |2 = ||2 + | |2 , where ||2 = b r 1 , c 1(b r 1 ) .

2.2. Stochastic discount factors. The usual way of obtaining stochastic discount factors in con-tinuous time is through risk-neutral measures. The Fundamental Theorem of Asset Pricing in the

present Ito process setting states that absence of Free Lunches with Vanishing Risk4

is equivalent tothe existence of a probability Q P such that S i is (only) a local Q -martingale for all i = 0 , . . . , d .(For the denition of local martingales, check for example [7].) In that case, by dening Y viaY t = t (dQ / dP )|F t , Y S i is a local P -martingale for all i = 0 , . . . , d . The last property will be takenhere as the denition of a stochastic discount factor.

Denition 2.3. Consider the above It o process set-up. A stochastic process Y will be called astochastic discount factor if

Y 0 = 1 and Y T > 0, P -almost surely. Y S i is a local P -martingale for all i = 0 , 1, . . . , d .

In the case where Y S 0 is an actual martingale, i.e., E P [Y T S 0T ] = S 00 , a risk-neutral measure Q isreadily dened via the recipe d Q = ( Y T S 0T /S 00 )dP . However, this is not always the case, as Example2.4 below will show. Therefore, existence of a stochastic discount factor is a weaker notion thanexistence of a risk-neutral measure. For some practical applications though, these differences areunimportant. There is further discussion of this point at 2.5 later.

Example 2.4. Let S 0 1 and S 1 be a 3-dimensional Bessel process with S 10 = 1. If F is the naturalltration of S 1, it can be shown that the only stochastic discount factor is Y = 1 /S 1, which is astrict local martingale in the terminology of [ 4].

2.3. Credit constraints on investment. In view of the theoretical possibility of continuoustrading, in order to avoid so-called doubling strategies (and in order to have the FundamentalTheorem of Asset Pricing holding), credit constraints have to be introduced. The wealth of agentshas to be bounded from below by some constant, representing the credit limit. Shifting the wealth

4Free Lunches with Vanishing Risk is the suitable generalization of the notion of Arbitrages in order to get aversion of the Fundamental Theorem of Asset Pricing in continuous time. The reader is referred to [3].

7/27/2019 1001.1184

9/12


appropriately, one can assume that the credit limit is set to zero; therefore, only positive wealthprocesses are allowed in the market.

Since only strictly positive processes are considered, it is more convenient to work with proportions of investment, rather than absolute quantities as was the case in Section 1. Pick some F -adaptedprocess = ( 1, . . . , d ). For i = 1 , . . . , d and t [0, T ], the number it represents the percentage of capital in-hand invested in asset i at time t. In that case, 0 = 1 di=1

i will be invested in thesavings account. Denote by X the wealth generated by starting from unit initial capital ( X 0 = 1)and invest according to . Then,

(2.1)dX tX t

=d

i=0 it

dS itS it

= ( r t + t , bt r t 1 ) d t + t t , dW t .

To ensure that the above wealth process is well-dened, we must assume that

(2.2) T

0| t , bt r t 1 |dt < + and

T

0 t , ct t dt < + , P -a.s.

The set of all d-dimensional, F -adapted processes that satisfy ( 2.2) is denoted by . A simpleuse of the integration-by-parts formula gives the following result.

Proposition 2.5. If Y is a stochastic discount factor, then Y X is a local martingale for all .

2.4. Connection with no free lunch notions. The next line of business is to obtain an exis-tential result about stochastic discount factors in the present setting, also connecting their existenceto a no-arbitrage-type notion. Remember from 1.4 the special stochastic discount factor that isthe reciprocal of the log-optimal wealth process. We proceed somewhat heuristically to computethe analogous processes for the It o-process model. The linear stochastic differential equation ( 2.1)has the following solution, expressed in logarithmic terms:

(2.3) log X =

0r t + t , bt r t 1

12

t , ct t dt +

0t t , dW t

Assuming that the local martingale term

0 t t , dW t in (2.3) is an actual martingale, the aimis to maximize the expectation of the drift term. Notice that we can actually maximize the driftpathwise if we choose the portfolio = c 1(b r 1 ). We need to ensure that is in . If is easyto see that ( 2.2) are both satised if and only if

T 0 |

t |2dt < P -a.s., where := c 1(b r 1 ) isthe special risk-premium of Proposition 2.2. Under this assumption, . Call X = X anddene

(2.4) Y :=1

X = exp

0t , dW t

12

0|t |2dt .

Use the integration-by-parts formula it is rather straightforward to check that Y is a stochasticdiscount factor. In fact, the ability to dene Y is the way to establish that a stochastic discountfactor exists, as the next result shows.

Theorem 2.6. For the It o process model considered above, the following are equivalent.

7/27/2019 1001.1184

10/12


(1) The set of stochastic discount factors is non-empty.(2)

T 0 |

t |2dt , P -a.s.; in that case, Y dened in (2.4) is a stochastic discount factor.(3) For any > 0, there exists = () R + such that P [X T > ] < uniformly over all

portfolios .

The property of the market described in statement (3) of the above Theorem is coined No Unbounded Prot with Bounded Risk in [6], where the interest reader is referred to.

The next structural result about the stochastic discount factors in the It o process setting revelsthe importance of Y as a building block.

Theorem 2.7. Assume that F is the ltration generated by the Brownian motion W . Then, any stochastic discount factor Y in the previous It o process model can be decomposed as Y = Y N ,where Y was dened in (2.4) and

N t = exp

t

0u , dW u

t

0|u |2du , t [0, T ]

where is a m-dimensional F -adapted process with = 0 .

If the assumption that F is generated by W is removed, one still obtains a similar result withN being replaced by any positive F -martingale N with N 0 = 1 that is strongly orthogonal toW . The specic representation obtained in Theorem 2.7 comes from the Martingale Representation Theorem of Brownian ltrations; check for example [7].

2.5. Stochastic discount factors and equivalent martingale measures. Consider an agentthat uses a stochastic discount factor Y for valuation purposes. There is a possibility that Y S i

could be a strict local P -martingale for some i = 0 , . . . , d , which would mean that 5 S i0 > EP [Y T S iT ].

The last inequality is puzzling in the sense that the agents indifference price for the ith asset,which is E P [Y T S iT ], is strictly lower than the market price S

i0. In such a case, the agent would be

expected to wish to short some units of the ith asset. This is indeed what is happening; however,because of credit constraints, this strategy is infeasible. Example 2.8 below will convince you of this fact. Before presenting the example, an important issue should be claried. One would rush tostate that such inconsistencies are tied to the notion of a stochastic discount factor as it appearsin Denition 2.3, and that is is strictly weaker than existence of a probability Q P that makes alldiscounted processes S i local Q -martingales for i = 0 , . . . , d . Even if such a probability did exist,S i could be a strict local Q -martingale for some i = 1 , . . . , d ; in that case, S i0 > E

Q [ T S iT ] and thesame mispricing problem pertains.

Example 2.8. Let S 0 1, S 1 be the reciprocal of a 3-dimensional Bessel process starting at S 10 = 1under P and F be the ltration generated by S 1. Here, P is the only equivalent local martingalemeasure and 1 = S 10 > E

P [S 1T ] for all T > 0. This is a complete market an agent can startwith capital E P [S 1T ] and invest in a way so that at time T the wealth generated is exactly S T .

5The inequality follows because positive local martingales are supermartingales see for example [ 7].

7/27/2019 1001.1184

11/12


Naturally, the agent would like to long as much as possible from this replicating portfolio and shortas much as possible from the actual asset. However, in doing so, the possible downside risk isinnite throughout the life of the investment and the enforced credit constraints will disallow forsuch strategies.

In the context of Example 2.8, the law of one price fails, since the asset that provides payoff S 1T at time T has a market price S 10 and a replication price E

P [S 1T ] < S 10 . Therefore, if the law of oneprice is to be valid in the market, one has to insist on existence of an equivalent ( true ) martingalemeasure Q , where each discounted process S i is a true (and not only local) Q -martingale forall i = 0 , . . . , d . For pricing purposes then, it makes sense to ask that the stochastic discountfactor Y that is chosen according to Theorem 2.7 is such that Y S i is a true P -martingale for alli = 0 , . . . , d . Such stochastic discount factors give rise to probabilities Q that make all deatedasset-price-process Q -martingales and can be used as pricing measures.

Let us specialize now to a Markovian model where the interest rate process ( r t )t[0,T ] is deter-

ministic and the process is a funnction of the assets, i.e., = ( S ), where is a deterministicfunction from R d to the space of ( m d)-matrices. As long as a claim written only on the tradedassets is concerned, the choice of Q for pricing is irrelevant, since the asset prices under Q havedynamics

dS itS it

= r t dt + i (S t ), dW t , t [0, T ], i = 1 , . . . , d ,

where W is a Q -Brownian motion; in particular, the process ( S )t[0,T ] has the same law underany Q . However, if one is interested on pricing a claim written on a non-traded asset whose priceprocess Z has P -dynamics

dZ t = a t dt + f t , dW t , t [0, T ]for F -adapted a and f = ( f 1, . . . , f m ), then the Q -dynamics of Z are

dZ t = ( a t f t , t f t , t ) d t + f t , dW t , t [0, T ]

The dynamics of Z will be independent of the choice of only if the volatility structure of theprocess Z , given by f , is in the range of = ( S ) . This will mean that f, = 0 for all such that = 0 and that Z is perfectly replicable using the traded assets. As long st there isany randomness in the movement in Z that cannot be captured by investing in the traded assets,i.e., if there exists some with = 0 and f, not being identically zero, perfect replicabilityfails and pricing becomes a more complicated issue, depending on the preferences of the particularagent as given by the choice of to form the stochastic discount factor.

References

[1] J. H. Cochrane , Asset Pricing , Princeton University Press, January 2001.[2] M. H. A. Davis , Option pricing in incomplete markets , in Mathematics of derivative securities (Cambridge,

1995), vol. 15 of Publ. Newton Inst., Cambridge Univ. Press, Cambridge, 1997, pp. 216226.

7/27/2019 1001.1184

12/12


[3] F. Delbaen and W. Schachermayer , The mathematics of arbitrage , Springer Finance, Springer-Verlag, Berlin,2006.

[4] K. D. Elworthy, X.-M. Li, and M. Yor , The importance of strictly local martingales; applications to radial Ornstein-Uhlenbeck processes , Probab. Theory Related Fields, 115 (1999), pp. 325355.

[5] H. F ollmer and A. Schied , Stochastic nance , vol. 27 of de Gruyter Studies in Mathematics, Walter deGruyter & Co., Berlin, extended ed., 2004.

[6] I. Karatzas and C. Kardaras , The numeraire portfolio in semimartingale nancial models , Finance Stoch.,11 (2007), pp. 447493.

[7] I. Karatzas and S. E. Shreve , Brownian motion and stochastic calculus , vol. 113 of Graduate Texts inMathematics, Springer-Verlag, New York, second ed., 1991.

[8] , Methods of mathematical nance , vol. 39 of Applications of Mathematics (New York), Springer-Verlag,New York, 1998.

[9] D. Lamberton and B. Lapeyre , Introduction to stochastic calculus applied to nance , Chapman & Hall,London, 1996. Translated from the 1991 French original by Nicolas Rabeau and Fran cois Mantion.

[10] S. A. Ross , The arbitrage theory of capital asset pricing , J. Econom. Theory, 13 (1976), pp. 341360.[11] J. von Neumann and O. Morgenstern , Theory of games and economic behavior , Princeton University Press,

Princeton, NJ, anniversary ed., 2007. With an introduction by Harold W. Kuhn and an afterword by ArielRubinstein.

Constantinos Kardaras, Mathematics and Statistics Department, Boston University, 111 Cumming-

ton Street, Boston, MA 02215, USA.

E-mail address : [email protected]

1001.1184

Documents

Transcript of 1001.1184